Let w be the vorticity of a stationary solution of the two-dimensional Navier-Stokes equations with a drift term parallel to the boundary in the half-plane -infty1, with zero Dirichlet boundary conditions at y=1 and at infinity, and with a small force term of compact support. Then, |xyw(x,y)| is uniformly bounded in the half-plane. The proof is given in a specially adapted functional framework and complements previous work.